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PAPER IV: NUMERICAL METHODS

Unit I
: SOLUTION OF NONLINEAR EQUATIONS: Newton’s method – Convergence of Newton’s method – Fixed point iteration: x=g(x) method – Bairstow”s Method for quadratic factors NUMERICAL DIFFERENTIATION AND INTEGRATION: Derivatives from Differences tables – Higher order derivatives – Divided difference, Central-Difference formulas – Composite formula of Trapezoidal rule – Romberg integration – Simpson’s rules.

Unit II
: SOLUTION OF SYSTEM OF EQUATIONS: The Elimination method – Gauss and Gauss Jordan methods – LU Decomposition method – Matrix inversion by Gauss-Jordan method – Methods of Iteration – Jacobi and Gauss Seidal Iteration – Relaxation method – Systems of Nonlinear equations.

Unit III
: SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS: Taylor series method – Euler and Modified Euler methods – Runge-Kutta methods – Multistep methods – Milne’s method – Adams Moulton method –Convergence Criteria – Systems of equations and Higher Order equations.

Unit IV
: BOUNDARY VALUE PROBLEMS AND CHARACTERISTIC VALUE PROBLEMS: The shooting method – solution through a set of equations – Derivative boundary conditions – Characteristic value problems – Eigen values of a matrix by Iteration – The power method.

Unit V
: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS: (Solutions of Elliptic, Parabolic and Hyperbolic partial differential equations) Representation as a difference equation – Solving for the temperatures in a Slab – Iterative methods – The Poisson equation – Derivative boundary conditions – Solving the equation for time-dependent heat flow (i) The Explicit method (ii) The Crank Nicolson method – Parabolic Equations in Two or Three dimensions – The wave equations in two dimensions..

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